CONTROL OF COOPERATIVE UNMANNED AERIAL VEHICLES A Thesis submitted for the degree of Doctor of Philosophy (Ph. D) by Kostas ALEXIS Dipl. ELECTRICAL & COMPUTER ENGINEERING UNIVERSITY OF PATRAS DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING THESIS NO: 277 JULY 2011 The following dissertation by Kostas Alexis, Dipl. Electrical & Computer Engineering, is approved: “Control of Cooperative Unmanned Aerial Vehicles” The dissertation was presented in public on July 6th . The seven member committee: 1.Anthony Tzes, Professor at Electrical & Computer Engineering Department, University of Patras, Member of the advisor committee 2.

Konstantinos Eustathiou, Assistant Professor at Electrical & Computer Engineering Department, University of Patras, Member of the advisor committee 3. Kimon Valavanis, Professor at Electrical & Computer Engineering Department, University of Denver, Member of the advisor committee 4. Stamatios Manesis, Associate Professor at Electrical & Computer Engineering Department, University of Patras 5. Stauros Koubias, Professor at Electrical & Computer Engineering Department, University of Patras 6.

Evangelos Dermatas, Assistant Professor at Electrical & Computer Engineering Department, University of Patras 7. Nikolaos Aspragathos, Professor at Mechanical Engineering & Aeronautics, University of Patras July 2011, Patras, Greece The Supervisor and Head of Electrical and Computer Engineering Department, University of Patras: Anthony Tzes, Professor ii Abstract This thesis addresses the problems of design and control of small cooperative unmanned autonomous quadrotor aerial vehicles. A new approach for the modeling of the system’s dynamics using linearized Piecewise Af? e Models, is proposed. The Piecewise Af? ne dynamic–models cover a large part of the quadrotor’s ? ight envelope while also taking into account the additive effects of environmental disturbances. The effects of aerodynamic forces and moments were also examined. A small quadrotor is designed and developed that emphasizes in the areas of increased on–board computational capabilities, state estimation and modular connectivity.

Based on the translational and rotational system’s dynamics: a) a switching model predictive controller, b) an explicitly solved constrained ? ite time optimal control strategy, and c) a cascade control scheme comprised of classical Proportional Integral Derivative control scheme augmented with angular acceleration feedback, were designed and experimentally tested in order to achieve trajectory tracking under the presence of wind–gusts. The ef? ciency of the proposed control methods was veri? ed through extended experimental studies. The ? nal quadrotor design utilizes a powerful control unit, a sensor system that provides state estimation based on inertial sensors, ultrasound sonars, GPS and vision chips, and an ef? cient actuating system.The research effort extended in the ? eld of unmanned aerial vehicles cooperation. Cooperation strategies were proposed in order to address the problems of: a) Forest Fire Monitoring and b) Unknown Area Exploration and Target Acquisition.

The Forest Fire Monitoring algorithm is formulated based on consensus systems theory formulated as a spatiotemporal rendezvous problem in between the quadrotors. The Area Exploration and Target Acquisition algorithm is formulated based on market–based approaches. iv This dissertation is dedicated to my family, my mother, father, and brother for their continuous love and supportAcknowledgements During the last four years and during my research process, i was helped and encouraged from several people. First of all I would like to thank my advisor, Prof. A. Tzes for his help and scienti? c guidance. His support and his trust in my research all these years helped me accomplish my goals and complete this arduous task. Especially, I would also like to thank Prof.

K. Valavanis for believing in my work, supporting me in order to gain a worldwide research perspective and aiding me in order to successfully claim funding.His pioneering work in the ? eld of Unmanned Aerial Vehicles inspired me during all these years. Additionally, I would like to thank Prof. K. Eustathiou for his research guidance and technical advice. Finally, i would like to thank Prof.

N. Aspragathos, Prof. E.

Dermatas, Prof. S. Koumpias and Prof. S.

Manesis. Their comments aided me substantially in the improvement of this dissertation. I would like to express my respect to my colleagues in our Lab for their continuous support. I have to give my special thanks to some of these people. Speci? ally (alphabetically): • Kostas Andrianesis for all his help, for all these times that i asked him to review my work and give me his helpful opinion.

I am sure you will ? nd your way based on your genuine work and the fact that you always focus on details. • Dr. Yiannis Koveos for all his technical help in order to design the required experimental set–ups. I hope you can ? nd the laboratory that is worth your dexterity.

• Prof. George Nikolakopoulos for helping me in my initial steps and for his cooperation during all these years. I am sure that you will be ok! Greetings to Sweden! Christos Papachristos, a new colleague but an old friend. He came at the right time with his passion and positive energy. I have to thank him for his cooperation as a colleague and his support as a friend (for all the other subjects that also affected my life). Keep up the good work and the future is yours! • Dr. Marialena Vagia for supporting me and giving helpful advice for my next feel comfortably. The success in your academic career is guaranteed based on your efforts! I would also like to thank all the other colleagues in the lab.

We passed joyful moments! The last MED conference was the peak of those.Always let Giannakis be the “coordinator”! Always trust Helen–she always organize everything. Apart from my advisors and colleagues i would like to thank my friends. Their companion, their advice, their different point of view, all the things that we made and lived together, all the moments of enjoyment, all the circumstances we faced, all these we fought for, all these moments play an important role in my life, a dominant role in the shaping of my personality. I thank them for being supportive. I would like to name some of them: Maria, Mitsos (Agori), Sterios (Gouky), Dimitris (Mitsarionas), Aggeliki, Katerina, Nikos – E? Kokonas, Giannakos, Aggelos – Ntina, Kanellis, Podias & Goldy, Mpountas, Stamatopoulos, Poluna, Stavros, Thanasis, Menegis, Boudas, Apostolou, Vassos, Sakellariou, Xristodoulou, Ligka, Thymios, Mpagkeris, Koukios, Alexopoulos T.

(Takaros), Alexopoulos L. , Thodwros Mplikas, Vakalop, Tsakalos, Tony, Tserolas, Dritsas. All these people (but not only them) played an important role in my life.

I wish you all the best! Finally, I would like to express my gratitude to my family, my mother, father and brother. They have helped me by all means. They were always part of the most important moments of my life and always encouraged me.I am aware of their struggle, of the ? nancial costs of my studies. I hope today they believe that their efforts were meaningful. steps.

Her support always encouraged me. Her personality always made me iv Contents List of Figures List of Tables 1 Introduction 1. 1 Scienti? c and Socioeconomic Motivation . . . .

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3 Unmanned Aerial Vehicles History . . .

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. . . ix xv 1 1 1 2 3 6 6 6 8 8Mini and Micro UAV Projects–State of the Art . .

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3. 3 1. 3. 4 Modeling .

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Cooperation . . . .

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2 2. 3 2. 3. 1 2. 3. 2 Quadrotor Dynamics . .

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12 General Forces and Moments . . . . . . . .

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15 Equations of Motion . . . . . . .

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22 Blade Flapping . . .

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. . . . . 26 Piecewise Af? ne Modeling Approach . . . . . . . . . . . . . . . . . . . . . . 18 Aerodynamics Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 v CONTENTS 2. 3. 3 2. 4 2. 5 Air? ow Disruption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Quadrotor Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 29 3 Quadrotor Design 3. 1 3. 1. 1 3. 1. 2 3. 2 3. 2. 1 3. 2. 2 3. 2. 3 3. 2. 4 Main Control Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ARM MCU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 pITX MCU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Attitude–Heading Reference System . . . . . . . . . . . . . . . . . . . 32 Xsens MTi-G AHRS . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Other IMUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Translational Motion Estimation . . . . . . . . . . . . . . . . . . . . . 34 3. 2. 4. 1 3. 2. 4. 2 3. 2. 5 Altitude Sonar . . . . . . . . . . . . . . . . . . . . . . . . . 34 Horizontal Optic Flow . . . . . . . . . . . . . . . . . . . . . 35 Sensor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Advanced Image Processing . . . . . . . . . . . . . . . . . . . . . . 38 Radio Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Telemetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Wi-Fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3. 3 Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3. 3. 1 3. 3. 2 3. 3. 3 3. 4 3. 5 Propulsion System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 UPATcopter(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3. 5. 1 3. 5. 2 3. 5. 3 3. 5. 4 Phase 1: Initial Design – Proof of Concept . . . . . . . . . . . . . . . 45 Phase 2: Advanced Actuation, Sensor and Computation UPATcopter . . 46 Phase 3: UPATcopter Final Design . . . . . . . . . . . . . . . . . . . . 47 Micro–UPATcopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3. 6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 51 4 Quadrotor Control 4. 1 4. 1. 1 4. 1. 2 Constrained Finite Time Optimal Control . . . . . . . . . . . . . . . . . . . . 52 CFTOC Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . 55 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4. 1. 2. 1 4. . 2. 2 Rotor Response Measurements . . . . . . . . . . . . . . . . 61 Quadrotor’s Regulation Response . . . . . . . . . . . . . . . 65 vi CONTENTS 4. 1. 2. 3 4. 1. 2. 4 4. 1. 2. 5 4. 1. 3 LQ and CFTOC Quadrotor response comparison . . . . . . . 67 Quadrotor’s Set–point Response . . . . . . . . . . . . . . . 69 Quadrotor’s Regulation Response under Wind Gusts with different directions . . . . . . . . . . . . . . . . . . . . . . . 74 Quadrotor Attitude CFTOC Experimental Results . . . . . . . . . . . . 77 4. 1. 3. 1 4. 1. 3. 2 Quadrotor’s CFTOC Regulation Response . . . . . . . . . . 79 CFTOC for set-point at roll angle . . . . . . . . . . . . . . 83 Effect of Sampling Time on Stability and Performance . . . . 86 4. 1. 4 4. 2 CFTOC for set-point at yaw angle . . . . . . . . . . . . . . . . . . . . 84 4. 1. 4. 1 Model Predictive Quadrotor Control . . . . . . . . . . . . . . . . . . . . . . . 88 4. 2. 1 4. 2. 2 4. 2. 3 4. 2. 4 Attitude/Translational MPC Quadrotor Control . . . . . . . . . . . . . 88 MPC Attitude/Translational Simulation Studies . . . . . . . . . . . . . 94 Experimental MPC Attitude/Translational Studies . . . . . . . . . . . . 97 Quadrotor MPC Attitude Control . . . . . . . . . . . . . . . . . . . . 104 4. 2. 4. 4. 2. 4. 2 4. 2. 4. 3 4. 2. 4. 4 Experimental Studies . . . . . . . . . . . . . . . . . . . . . 106 Attitude Regulation and Setpoint SMPC Quadrotor Response 108 Attitude Tracking SMPC Quadrotor Response . . . . . . . . 109 Attitude Regulation of a SMPC free-? ying Quadrotor at High Sampling Rate . . . . . . . . . . . . . . . . . . . . . . . . . 114 4. 3 PID/PIDD Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4. 3. 1 4. 3. 2 4. 3. 3 Attitude Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Altitude and Position Controllers . . . . . . . . . . . . . . . . . . . 122 Quadrotor PID/PIDD Controlled Experimental Studies . . . . . . . . . 122 4. 4 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 127 Cooperation of Unmanned Aerial Vehicles 5. 1 5. 1. 1 5. 1. 2 5. 1. 3 5. 1. 4 Cooperation of UAVs for Forest–Fire Monitoring . . . . . . . . . . . . . . . . 128 Cooperative Strategy Theory Elements . . . . . . . . . . . . . . . . . 129 Forest Fire Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5. 1. 2. 1 5. 1. 3. 1 Forest Fire Propagation . . . . . . . . . . . . . . . . . . . . 130 Problem Formulation . . . . . . . . . . . . . . . . . . . . 132 Quadrotor Rendezvous for Forest Fire Surveillance . . . . . . . . . . . 131 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 vii CONTENTS 5. 2 Cooperation of UAVs for Area Coverage and Target Acquisition . . . . . . . . 139 5. 2. 1 5. 2. 2 Heterogenous UAV Cooperation for Area Exploration and Target Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5. 3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 147 6 General Conclusions and Future Work 6. 6. 2 6. 3 6. 4 6. 5 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 UPATcopter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Quadrotor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 151 155 7 Curriculum Vit? References viii List of Figures 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 Older and new designs of UAVs and quadrotor helicopters . . . . . . . . . . Possible UAV applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . European Civil and Commercial UAV Market . . . . . . . . . . . . . . . . . . Predicted progress of the UAV markets . . . . . . . . . . . . . . . . . . . . . . UAV anticipated short term activity . . . . . . . . . . . . . . . . . . . . . . . . Unmanned Aerial Vehicles projects State of the Art . . . . . . . . . . . . . . . UPATcopter: The experimental quadrotor prototype developed in University of Patras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. 1 2. 2 2. 3 2. 4 2. 5 2. 6 2. 7 2. 8 3. 3. 2 3. 3 3. 4 3. 5 9 2 4 4 5 5 7 Visualization of the main dynamic principles of the quadrotor rotorcraft . . . . 11 Quadrotor helicopter con? guration frame system . . . . . . . . . . . . . . . . 12 Induced air? ow velocity reduction and rotor tip vortex reduction due to ground effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Subsystems of quadrotor’s dynamics . . . . . . . . . . . . . . . . . . . . . . . 20 Effective Lift in hovering and in forward motion . . . . . . . . . . . . . . . . . 24 Effect of blade ? apping due to wind–gusts or forward motion . . . . . . . . . 25 Blade Flapping Angle Rotation and relative experiments using a dc–brushless motor (b) and a dc–brushed motor (c) . . . . . . . . . . . . . . . . . . . . . . 25 UPATcopter Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Luminary Micro ARM7 Main Control Unit . . . . . . . . . . . . . . . . . . . 30 pITX Main Control Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Xsens Mti-G IMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ArduIMU, Ardupilot’s Sensor board and Sparkfun Razor IMU . . . . . . . . . 34 Mouse Sensor–based Optic Flow device . . . . . . . . . . . . . . . . . . . . 36 ix LIST OF FIGURES 3. 6 3. 7 3. 8 3. 9 Tam2 Vision Chip combined with an Arduino microprocessor . . . . . . . . . 36 Integrated indoors sensor system for quadrotor state estimation . . . . . . . . . 38 SRV–1 Black? n Camera module . . . . . . . . . . . . . . . . . . . . . . . . . 39 RC and Joystick remote control options . . . . . . . . . . . . . . . . . . . . . 40 3. 10 BL–CTRL ESC’s left NMOS, right PMOS thermal characteristics . . . . . . . 42 3. 11 Selected Electronic Speed Controller, Motor and Propellers . . . . . . . . . . . 43 3. 12 Experimental set–up used for motor–propeller characteristics measurements . 43 3. 13 Experimental ESC–dc bushless motor–propeller system response . . . . . . . . 44 3. 14 3D–CAD UPATcopter initial design 3D Cad . . . . . . . . . . . . . . . . . . . 45 3. 15 Initial UPATcopter Design attached at Heli–stand . . . . . . . . . . . . . . . . 45 3. 16 Structure of the second design of the UPATcopter . . . . . . . . . . . . . . . . 46 3. 17 Second design of the UPATcopter . . . . . . . . . . . . . . . . . . . . . . . . 46 3. 18 Final design of the UPATcopter . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3. 19 UPATcopter main hardware diagram . . . . . . . . . . . . . . . . . . . . . . . 8 3. 20 Photos of the design and programming phases of UPATcopter . . . . . . . . . . 49 3. 21 Hovering UPATcopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3. 22 Micro–UPATcopter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4. 1 4. 2 4. 3 4. 4 4. 5 4. 6 Quadrotor CFTO–control Structure . . . . . . . . . . . . . . . . . . . . . . . . 53 UPATcopter simulation CFTOController ? –angle response . . . . . . . . . . . 56 UPATcopter simulation CFTOController ? –angle response . . . . . . . . . . . 56 UPATcopter CFTO-Controller partitioning . . . . . . . . . . . . . . . . . . . 56 Rotational rates disturbances caused by the wind–gust . . . . . . . . . . . . . . 57 UPATcopter simulation CFTOController ? –angle response under wind–gust disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4. 7 UPATcopter simulation CFTOController ? –angle response under wind–gust disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4. 8 4. 9 UPATcopter CFTOC and State Estimation . . . . . . . . . . . . . . . . . . . . 59 Quadrotor Attitude Control Experimental Set–Up . . . . . . . . . . . . . . . . 59 4. 10 Wind–Gust generating Set–Up . . . . . . . . . . . . . . . . . . . . . . . . . 60 4. 11 Reference motor voltage versus rotor’s angular velocity mapping for different wind–gust conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 x LIST OF FIGURES 4. 12 Rotor’s angular velocity versus Thrust and vertical Air? ow characteristic curves for different wind–gust conditions . . . . . . . . . . . . . . . . . . . . . . . . 63 4. 13 Air? ow studies using dry ice . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4. 14 Effect of blade ? apping due to wind–gusts . . . . . . . . . . . . . . . . . . . . 64 4. 15 Blade ? pping angle (a) and photo of a rotating at hovering speed propeller subject to a uni–directional wind–gust (b) . . . . . . . . . . . . . . . . . . . . 64 4. 16 Quadrotor’s (CFTOC regulation problem) roll, pitch, and yaw responses for s = 1(red), 3(green) and 5(blue), in the absence of wind gust . . . . . . . . . . 65 4. 17 Quadrotor’s (regulation problem) roll, pitch, and yaw responses for s = 1(red), 3(green) and 5(blue), subject to y-directional 3. 0m/s Wind Gust . . . . . . . . 66 4. 18 Attitude Regulation Responses for 5–PWA with wind gusts (unconstrained (blue) and constrained control (red)) . . . . . . . . . . . . . . . . . . . . . . 68 4. 19 Attitude Regulation Control Responses for 5–PWA with wind gusts (no constraints (blue) and constraints (red)) . . . . . . . . . . . . . . . . . . . . . . . 68 4. 20 LQ (red) & CFTOC (blue) rotation angles responses in absence of wind–gusts . 69 4. 21 LQ (red) & CFTOC (blue) control action responses in absence of wind–gusts . 70 4. 22 LQ (red) & CFTOC (blue) rotation angles responses under presence of wind– gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4. 23 LQ (red) & CFTOC (blue) control actions responses under presence of wind– gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4. 24 Comparison of Rotation Angles Setpoint Responses in the absence of wind gusts (1–PWA (red), 3–PWA (green), 5–PWA (blue)) . . . . . . . . . . . . . . 72 4. 25 Comparison of Rotation Angles Setpoint Responses in the presence of wind gusts (1–PWA (red), 3–PWA (green), 5–PWA (blue)) . . . . . . . . . . . . . . 73 4. 26 Comparison of Rotation Angles Setpoint Responses in the absence of wind gusts (unconstrained (blue) and constrained control (red)) . . . . . . . . . . . . 74 4. 27 Attitude Regulation Responses for 5–PWAs without wind gusts (unconstrained (blue) and constrained control (red)) . . . . . . . . . . . . . . . . . . . . . . 75 4. 28 Directions of the applied wind–gusts . . . . . . . . . . . . . . . . . . . . . . . 75 4. 29 CFTOC Roll Angles Responses (regulation) subject to directional Wind Gusts 4. 30 CFTOC Pitch Angles Responses (regulation) subject to directional Wind Gusts 76 76 4. 31 CFTOC Yaw Angles Responses (regulation) subject to directional Wind Gusts . 77 xi LIST OF FIGURES 4. 32 CFTOC Regulation State Responses and Control Effort for the case of 3-PWA Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4. 33 Quadrotor’s roll (? ), pitch (? ) and yaw (? responses for s = 1, 3, 5 in the absence of wind gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4. 34 CFTOC Comparison of Rotation Angles Responses subject to y-directional 3. 0m/s Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4. 35 CFTOC Comparison of Rotation Angles Responses subject to a directional Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4. 36 CFTOC Comparison of Rotation Angles Setpoint (in the absence of wind–gusts) 83 4. 37 Comparison of Rotation Angles Setpoint Responses subject to a directional Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4. 38 Comparison of Rotation Angles Setpoint Responses in the absence of wind gusts 85 4. 39 Comparison of Rotation Angles Setpoint Responses subject to a directional Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4. 40 CFTOC Comparison of Rotation Angles Responses in the absence of wind–gusts 86 4. 41 CFTOC Comparison of Rotation Angles Responses subject to y-directional 3. 0m/s Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4. 42 CFTOC Comparison of Rotation Angles Responses subject to x(1. 8m/s), y(3. 86m/s) and z(1. 67m/s) directional Wind Gust . . . . . . . . . . . . . . . . 87 4. 43 Quadrotor Model Predictive Control Scheme . . . . . . . . . . . . . . . . . . 89 4. 44 Operating points for the set of utilized PWA systems . . . . . . . . . . . . . . 94 4. 45 Quadrotor MPC cyclic path response . . . . . . . . . . . . . . . . . . . . . . . 95 4. 46 Quadrotor MPC Ex , Ey , Ez responses for cyclic path . . . . . . . . . . . . . . . 96 4. 47 Quadrotor MPC boustrophedon path response . . . . . . . . . . . . . . . . . . 96 4. 48 Quadrotor MPC Ex , Ey , Ez responses for boustrophedon reference path . . . . 96 4. 9 Quadrotor MPC Flight path position hold and altitude set–point tracking . . . . 98 4. 50 Quadrotor’s estimated x, y horizontal position, optic ? ow measurements and altitude data of the ? ight path (position hold and altitude set–point tracking) . . 99 4. 51 Quadrotor’s MPC Flight path in setpoint altitude/hold–position control under the in? uence of wind–gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4. 52 Quadrotor MPC estimated x, y horizontal position, optic ? ow measurements and altitude data of the ? ight path under the presence of a directional wind gust 100 4. 53 Experimental quadrotor MPC ? ght path for translational set–point trajectory . 101 xii LIST OF FIGURES 4. 54 Quadrotor’s estimated x, y, z position ? ight path of the quadrotor’s MPC ? ight for translational set–point trajectory . . . . . . . . . . . . . . . . . . . . . . . 101 4. 55 Quadrotor MPC altitude and attitude response for hovering trajectory in the absence of wind disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4. 56 Quadrotor MPC altitude and attitude response for hovering trajectory under the presence of forcible wind–gusts . . . . . . . . . . . . . . . . . . . . . . . . . 102 4. 57 Quadrotor MPC attitude regulation response . . . . . . . . . . . . . . . . . . 103 4. 58 Quadrotor’s MPC Roll, pitch and yaw rates and PWA switching during the attitude regulation response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4. 59 Switched Model Predictive Controller/Identi? cation Structure . . . . . . . . . 109 4. 60 Quadrotor’s MPC attitude regulation response . . . . . . . . . . . . . . . . . . 110 4. 61 Quadrotor’s MPC attitude setpoint response . . . . . . . . . . . . . . . . . . . 110 4. 62 Quadrotor’s MPC attitude response for the pulse-wave tracking problem in the absence of wind gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4. 63 Quadrotor’s MPC attitude response for the pulse-wave tracking problem inside a directional wind gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4. 64 Quadrotor’s MPC attitude response for sinus reference inputs in the absence of wind gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4. 65 Quadrotor’s MPC attitude response for sinus reference inputs inside a directional wind gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4. 66 Quadrotor’s MPC attitude response for sinusoidal reference inputs in the absence of wind gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4. 67 Quadrotor’s MPC attitude response for the case of sinusoidal references inside a x(1. 31m/s), y(3. 84m/s) and z(1. 65m/s) directional wind gust . . . . . . . . . 115 4. 68 Quadrotor’s MPC attitude response for the case of sinusoidal references at the presence of a 10 sec sudden x(1. 8m/s), y(4. 02m/s) and z(1. 9m/s) directional wind gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4. 69 Quadrotor SMPC Performance w. r. t. various p and m Parameters . . . . . . . . 116 4. 70 Free-Flying Quadrotor Attitude MPC Regulation Response for Ts = 0. 1sec . . 117 4. 71 Free-? ying Quadrotor Attitude MPC Regulation Response for Ts = 0. 01sec subject to a directional Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . 117 4. 72 Quadrotor/Actuator Dynamics Interconnection . . . . . . . . . . . . . . . . . 118 4. 73 PID/PIDD Position/Attitude Control Scheme . . . . . . . . . . . . . . . . . . 119 xiii LIST OF FIGURES 4. 74 Quadrotor PID/PIDD Position/Attitude Control Frequency and Step Responses 120 4. 75 Quadrotor PIDD/PID step response using [different](optimized) PIDD gains [left](right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4. 76 Quadrotor PIDD/PID roll and yaw step responses for various gains . . . . . . . 121 4. 77 Quadrotor PID/PIDD Control measured ? ight path in Ex Ey axis in position hold experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4. 78 Quadrotor PID/PIDD Control measured x, y, z ? ight path in position hold experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4. 79 Quadrotor PID/PIDD attitude regulation experiments using PIDD controllers . 125 4. 80 Quadrotor PID/PIDD Control aggressive attitude regulation maneuver using PIDD controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5. 1 5. 2 5. 3 5. 4 5. 5 5. 6 5. 7 5. 8 5. 9 Cooperative UAVs for Mission Critical Issues . . . . . . . . . . . . . . . . . . 128 UAV–cooperation scheme using consensus algorithms . . . . . . . . . . . . . . 130 Forest Fire Propagation over a 6 hour interval (hourly updates) . . . . . . . . . 131 Weather conditions for Forest Fire Propagation . . . . . . . . . . . . . . . . . 132 Double Quadrotor cooperation for forest ? re surveillance . . . . . . . . . . . . 133 Rendezvous Study for an Evolving Fire Perimeter (case (a) thru (c)) . . . . . 135 Rendezvous Study for an Evolving Fire Perimeter (case (d) thru (f)) . . . . . . 136 Rendezvous Study for an Evolving Fire Perimeter (case (g) thru (h)) . . . . . . 137 Rendezvous Study for an Evolving Fire Perimeter (case (i) thru (j)) . . . . . . . 137 5. 10 Rendezvous Study for a Folium–shaped Fire Perimeter . . . . . . . . . . . . . 138 5. 11 Representation of the Heterogenous UAV Area Exploration Strategy . . . . . . 140 5. 12 Omnidirectional ? ying camera and exploration camera visualizations . . . . . . 141 5. 13 Market–Based Area Exploration approach . . . . . . . . . . . . . . . . . . . . 142 5. 4 Area Exploration – Scenario A – ? nal snapshot . . . . . . . . . . . . . . . . . . 143 5. 15 Area Exploration – Scenario B – ? nal snapshot . . . . . . . . . . . . . . . . . . 144 5. 16 Area Exploration – Scenario C – a snapshot . . . . . . . . . . . . . . . . . . . . 145 xiv List of Tables 2. 1 2. 2 2. 3 2. 4 2. 5 2. 6 2. 7 3. 1 3. 2 3. 3 3. 4 3. 5 3. 6 4. 1 4. 2 4. 3 4. 4 4. 5 4. 6 4. 7 Rolling Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Pitching Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Yawing Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Forces Along Ez –axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Forces Along Ex –axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Forces Along Ey –axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Quadrotor Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Luminary–Micro Speci? cations . . . . . . . . . . . . . . . . . . . . . . . . . 30 pITX MCU Speci? cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 XSens MTi-G AHRS Speci? cations . . . . . . . . . . . . . . . . . . . . . . . 33 SRV–1 Black? n Camera Speci? cations . . . . . . . . . . . . . . . . . . . . . 38 Robbe 2827-35 dc–brushless motor characteristics . . . . . . . . . . . . . . . 41 Quadrotor Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Modi? ed Dragan? yer Quadrotor Parameter Values . . . . . . . . . . . . . . . 60 No. PWA vs. Linearization points vs. Guard Functions . . . . . . . . . . . . . 61 MSE for the CFTOC regulation problem, utilizing 1,3, and 5 PWAs with and without Wind Gust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 MSE for the CFTOC regulation problem subject to wind gusts for the unconstrained and constrained controller . . . . . . . . . . . . . . . . . . . . . . . 67 MSE for the CFTOC and LQ, utilizing 5 PWA models . . . . . . . . . . . . . 71 MSE for the set–point problem, utilizing 1,3, and 5 PWAs . . . . . . . . . . . 73 MSE for the CFTOC setpoint problem for the unconstrained and constrained controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 xv LIST OF TABLES 4. 8 4. 9 5. 1 5. 2 5. 3 Number of PWA systems Vs. CFTO Controller Regions . . . . . . . . . . . . 84 Quadrotor MPC PWA Operation Regions and Linearization Points (? ? 1) . . 98 Simulation Parameters for area exploration scenario (a) . . . . . . . . . . . . 143 Simulation Parameters for area exploration scenario (b) . . . . . . . . . . . . . 144 Simulation Parameters for area exploration scenario (c) . . . . . . . . . . . . . 145 xvi 1 Introduction A helicopter is a collection of vibrations held together by differential equations. 1. 1 Scienti? c and Socioeconomic Motivation 1. 1. 1 Unmanned Aerial Vehicles History Prototypes of Unmanned Aerial Vehicles (UAVs) have been developed in early 1900s (1). Several systems were developed for military operations during World Wars I & II, functioning as remote-controlled aircraft, mentioned as Remotely Piloted Vehicles (RPVs).The leap towards autonomous UAVs occurs in the second part of the 20th century. Advances in the areas of mechanics, computer science and automatic control led an evolution of aircraft–prototypes, with an enlarged list of possible applications, including wide-area event surveillance (e. g. wild? re breakout monitoring), hazardous conditions inspection services (such as power lines, collapsed or nuclear radiation-polluted buildings), agricultural services, aerial photography – mapping and reconnaissance, search and assist missions and many others. The target drone (Ryan Firebee) UAV developed in 1948 as a is shown in Figure 1. (part a) where its more recent (late 1900s) design Global hawk is presented to its right. Quadrotor is an aircraft that was ? rstly proposed by De Bothezat, whose design is shown in Figure 1. 1 (part b) in comparison with a current quadrotor UAV designed from Microdrones Gmbh. Currently UAV designs mainly focus in miniaturizing existing aircraft designs. Miniature-sized, power-ef? cient computers capable of not only handling tasks such as vehicle navigation, but of evaluating decisions, are being incorporated into Unmanned Systems and 1 1. INTRODUCTION (a) (b) Figure 1. : Older and new designs of UAVs and quadrotor helicopters combined with state-of-the-art environmental perception sensors. The development of environmental perception algorithms are giving birth to autonomous intelligent systems. As a result, the area of UAVs is populated by advanced platforms, capable of performing autonomous operations. The achievement of a maximum degree of ? ight autonomy is another major ? eld of scienti? c research, where the emphasis is in the design of advanced lightweight materials, novel UAV platform designs and renewable power sources. 1. 1. 2 Scienti? Motivation From the early days of Da Vinci and the Wright brothers, humanity has seen a great scienti? c and productivity breakthrough due to aviation. Important historical moments include fast intercontinental transport and market, the in? uence in combat–missions during the World Wars and the Space exploration. Recently, the research and production trends turned towards the development of large and small scale UAVs. Small and miniature aerial vehicles and particularly rotorcrafts pose signi? cant scienti? c and engineering problems that must be addressed in order to be able to ? autonomously and ef? ciently. These machines are characterized by very aggressive (fast) dynamics due to their low inertia moments and are subject to complex aerodynamic effects that affect their ? ight. 2 1. 1 Scienti? c and Socioeconomic Motivation This sets very strict requirements in their state estimation problem and controller implementation at high update rates. However, low–cost sensory systems are noisy and present drifting characteristics and thus the control problem becomes even more complex. Additionally, there are constraints in the utilized actuators.Computational power is another important issue since these systems must be able to perform onboard calculations for state estimation, ? ight control and environmental perception. Consequently, the problem of electro–mechanical design and autonomous control of these systems is very challenging. This problem becomes even more demanding if the effects of atmospheric disturbances are taken into account in order to develop systems able to navigate in typical mission environments. Finally, the cooperation of these systems is another important issue that has to be addressed through effective algorithms.Cooperative UAV–teams are sophisticated complex systems, whose control and autonomous navigation is an open scienti? c challenge. 1. 1. 3 Socioeconomic Motivation This research aims at the development of innovative intelligent and multifunctional autonomous cooperative unmanned aerial vehicles and specially quadrotors that will be able to be utilized in important real–life applications. Example scenarios of possible applications could be: a) area coverage, b) target acquisition, c) rescue and surveillance and many more as shown in Figure 1. 2.All these applications are addressing the threat of human lives and natural resources, for both the rescue or surveillance crews and the humans that are affected by the crisis. Research on unmanned aircrafts is of great importance also in means of economic and productivity growth. Current estimations and predictions on the European Civil and Commercial UAV Market are shwon in Figure 1. 3. Unmanned systems that would participate in such applications must be capable of autonomous ? ight under severe environmental conditions that are also capable of complex control, cooperation, environmental perception computations and modular connectivity.It should be noted that UAVs can be used in military and civilian applications. The potential of the civilian market is considerable larger that the military sector, although there are major constraints on this emerging market. At some point – once present obstacles have been overcome – it is expected that the civilian market will overtake the military market in value as it can be observed in the following projection shown in Figure 1. 4. 3 1. INTRODUCTION Figure 1. 2: Possible UAV applications Figure 1. 3: European Civil and Commercial UAV Market 1. 1 Scienti? c and Socioeconomic Motivation Figure 1. 4: Predicted progress of the UAV markets In the short–term, as shown in Figure 1. 5, it is likely that the majority of systems used will be Medium–Sized or small UAV systems (S/MUAS). Figure 1. 5: UAV anticipated short term activity In general, the offered research efforts will offer market chances for European and particularly Greece industry in promising areas, strengthening the national position in existing markets, such as UAVs, mechatronics, control systems and Intelligent systems.Moreover, Greece and Europe will reinforce the competitiveness of its industry in the growing UAV market. It is of paramount importance to mention at this point, that based on recent calculations the UAV marked exceeded the 5. 5B US dollars (only in U. S. A. ) in 2010, while several countries 5 1. INTRODUCTION are following the US lead and are likewise investing in UAVs. European countries are both embarking on their own acquisition programmes or working on collaborative projects. 1. 2 Mini and Micro UAV Projects–State of the Art This period is characterized by an exponential growth in advances in the area of UAVs.Research projects tackling the problems of control, autonomous navigation and cooperation of UAVs suddenly spread out in many universities and research centers around the world. These research entities use either commercial quadrotors and emphasize in the areas of cooperation (sFly, Flying Arena, MAST) or emphasize in the area of design and control of the aerial vehicle (PIXHAWK, STARMAC, Maple, DelFly, Hummingbird). In Figure 1. 6 some photos from the most interesting projects are presented. 1. 3 Contributions of this work This thesis focuses on design and control of cooperative quadrotors.The contribution of this work lies in the following ? elds: • New modeling approach of the quadrotor’s rotational and translational linearized dynamics. • New experimental platform design emphasizing in the areas of enhancing the onboard computational power and the sensor systems. • New system control approaches utilizing switching predictive control approaches and PID techniques augmented with angular acceleration feedback, emphasizing in the areas of constraints and disturbance effects modeling. • UAV cooperation: algorithms are proposed addressing the problems of forest ? e surveillance and area exploration. 1. 3. 1 Modeling A quadrotor is a highly nonlinear system with aggressive dynamics and a complicated set of aerodynamics effects that affect its ? ight. Within the framework of this research work a new limodel based on the theory of Piecewise Af? ne systems is proposed. The proposed modeling 6 1. 3 Contributions of this work Project Leading Research Group Picture sFly ETH Zurich Flying Arena ETH Zurich MAST UPENN STARMAC Stanford PIXHAWK ETH Zurich Maple U. Maryland DelFly II TU Delft Hummingbird Autonomous Helicopter Testbed Draganfly DARPA U. Denver GATECHFigure 1. 6: Unmanned Aerial Vehicles projects State of the Art 7 1. INTRODUCTION approach approximates the quadrotor’s nonlinear dynamics as a family of linearized Piecewise Af? ne systems and provides a better coverage of the vehicle’s ? ight envelope, compared to classical linearized approaches. Additionally, the proposed model also accounts for the effects of atmospheric disturbances as additive–af? ne terms. Additionally, the dynamics of the actuating system comprised of the electronic speed controller, the dc–brushless motor and the propeller were identi? ed and integrated in the system model.Special attention was also paid in deriving the mathematical formulation of the state input constraints that rule the quadrotor’s ? ight. 1. 3. 2 UPATcopter design An experimental quadrotor platform (UPATcopter) was developed as a part of this thesis. The UPATcopter is a powerful small sized autonomous aerial vehicle. Its design emphasizes in the areas of: a) powerful onboard computational capabilities, b) design of autonomous state estimation using sensor data fusion, c) extended connectivity options, and d) ef? cient actuators with relatively low power consumption. The ? al design of the UPATcopter utilizes a high–end processor as its Main Control Unit and multi–tasking operating system, an approach that provides the capability to implement sophisticated controllers, cooperation algorithms and image perception techniques in high–level programming languages. The developed sensory system consists of inertial sensors, ultrasound sonars, GPS (for outdoor ? ight) and vision sensors (for indoor ? ight). Through the implementation of nonlinear Extended Kalman Filter–based sensor fusion algorithms complete the state–vector autonomous estimation both outdoors and indoors was achieved.The developed actuation system’s performance is veri? ed through experimental studies and the control law of the speed controller was matched to the torque characteristics of the motor–propeller system. The UPATcopter is able to communicate usingseveral wireless protocols such as IEEE 802. 11 and 802. 15. 4. Its structure is made of carbon ? bers and anodized aluminium with the addition of dampeners in selected points in order to reduce the structural noise. An image of the ? nally developed UPATcopter is shown in Figure 1. 7. 1. 3. 3 Control The following control approaches were designed during this research. A Switching Model Predictive Control for the complete 6–Degrees of Freedom trajectory control of the quadrotor. This control approach was based on multiple Piecewise 8 1. 3 Contributions of this work Figure 1. 7: UPATcopter: The experimental quadrotor prototype developed in University of Patras Af? ne representations of the quadrotor’s attitude, altitude and horizontal dynamics and the additive effects of atmospheric disturbances. The developed controller utilizes elements from the model predictive control theory and takes into account the input and state constraints of the system.The proposed control ensures precise trajectory control, aggressive attitude maneuvering and effective attenuation of the effects induced by wind–gusts. • A Constrained Finite Time Optimal Control for the quadrotor’s attitude control. This approach is promising in terms of performance, while ensuring the stability during the switching among the different piecewise af? ne representations. Its drawback is due to its high computational costs that is needed fro the of? ine computation of the control commands. • A PIDD/PID control scheme for the attitude/translational dynamics of the system.This approach increased the bandwidth of the attitude controller due to angular acceleration feedback and also provided the capability to achieve position control. Extended experimental studies both in the absence and under the presence of forcible wind– gusts indicate the overall performance of the proposed control methods. 9 1. INTRODUCTION 1. 3. 4 Cooperation Additionally, cooperative strategies for unmanned aerial vehicles were also proposed. The proposed cooperation algorithms tackle the problems of a) forest ? re surveillance using a team of quadrotors and b) the problem of area exploration using a team of heterogenous aerial vehicles.The Forest Fire Monitoring algorithm is formulated based on consensus problems theory and speci? cally as a decentralized rendezvous in space between the members of the UAV team. The Area Exploration and Target Acquisition algorithm is formulated based on market–based approaches. 10 2 Quadrotor Modeling The quadrotor’s motion is governed by the lift forces produced by the rotating propeller blades, while the translational and rotational motions are achieved by means of difference in the counter rotating blades. Speci? ally: a) the forward motion is achieved by the difference in the lift force produced from the front and the rear rotors velocity, b) the sidewards motion by the difference in the lift force from the two lateral rotors, while c) the yaw motion is produced by the difference in the counter–torque between the two pairs of rotors front–right and back–left. Finally, motion at the perpendicular axis is produced by the total rotor thrust. The aforementioned basic principles of the quadrotor’s dynamics are visualized in Figure 2. 1. Figure 2. 1: Visualization of the main dynamic principles of the quadrotor rotorcraftThe main dynamic model of the quadrotor utilized in this thesis assumes that the structure 11 2. QUADROTOR MODELING of the craft is rigid and symmetrical, the center of gravity and the body ? xed frame origin coincide, the propellers are rigid and the thrust and drag forces are proportional to the square of propeller’s speed. Although these assumptions are revisited afterwards in order to take into account the aerodynamic effects, they are useful for the derivation of the main dynamic model of the system. Two coordinate systems have been utilized, 1) the Body–? xed frame (BFF) B = [B1 , B2 , B3 ]T and 2) the Earth–? ed frame (EFF) E = [Ex , Ey , Ez ]T as presented in Figure 2. 2. T4,M4 T1,M1 T3,M3 ? ? B3 T2,M2 B1 B2 ? Ex Ey Ez Figure 2. 2: Quadrotor helicopter con? guration frame system 2. 1 Quadrotor Dynamics In order to derive a precise representation of the Nonlinear model of the ? ying quadrotor the Newton–Euler Formulation has been followed. The Netwon–Euler equations describe the combined translational and rotational dynamics of a rigid body. According to this method, the dynamics of a rigid body under external forces applied to the Center of Mass (CoM) expressed in the Body–Fixed Frame take the following general form: ms I3? 0 0 I ? V ? ?V + ? ? ? ? I? F ? = (2. 1) ,where F, ? , I3? 3 , I, ms , V, ? correspond to total force vector acting on the CoM, total torque acting about the CoM, identity matrix, moment of inertia about the CoM, total mass of the body, acceleration of the CoM, and angular velocity of the body respectively. 12 2. 1 Quadrotor Dynamics Let RBFF>EFF be the square rotation matrix from BFF to EFF de? ned as: cos ? cos ? ? = ? cos ? sin ? ? sin ? ? sin ? sin ? cos ? ? cos ? sin ? sin ? sin ? sin ? + cos ? cos ? sin ? cos ? ? cos ? sin ? cos ? + sin ? sin ? ? . cos ? sin ? ? sin ? cos ? ? (2. 2) cos ? cos ? RBFF>EFFSpecial attention should be paid in the difference between the body rates p, q, r measured in BFF and the Tait-Bryan angle rates expressed in EFF: ? ? ? cos ? p ? ? ? ?q? = ? 0 r sin ? ?? ? ? 0 ? cos ? sin ? ? ?? ? ? 1 sin ? ? ?? ? . ? 0 cos ? cos ? ? (2. 3) The main aerodynamic forces and moments acting on the quadrotor, during a hovering ? ight segment, correspond to the thrust (T), the hub forces (H), and the drag moment (Q) due to vertical, horizontal and aerodynamic forces respectively, followed by the rolling moment (R) related to the integration, over the entire rotor, of the lift of each section, acting at a given radius.Based on the work presented at (2, 3) the aforementioned forces and moments take the following form: Thrust Force: is the resultant force of the vertical forces acting on all the blade elements = CT ? A(? Rrad )2 (2. 4) ? tw 1 1 1 CT = ( + µ 2 )? 0 ? (1 + µ 2 ) ? ?, ?? 6 4 8 4 where Rrad , ? , µ , ? , ? , ? , ? 0 , ? tw correspond to the rotor disc radius, solidity ratio, rotor advance ratio, lift slope, in? ow ratio, air density, pitch of incidence, and twist pitch respectively and CT represents the thrust coef? cient. Hub Force: is the resultant of the horizontal forces acting on all the blade elements T H = CH ?A(? Rrad )2 (2. 5) ? tw 1 ? 1 CH ), = µCd + ? µ (? 0 ? ?? 4 4 2 where CH represents the hub force coef? cient. Drag Moment: This moment about the rotor shaft is caused by the aerodynamic forces acting on the blade elements. The horizontal forces acting on the rotor are multiplied by the moment arm and integrated over the rotor. Drag moment determines the power required to spin the rotor. Q = CQ ? A(? Rrad )2 Rrad 1 1 1 1 CQ ? = (1 + µ 2 )Cd + ? ( ? 0 ? ?tw ? ? ) , ?? 8 6 8 4 (2. 6) 13 2. QUADROTOR MODELING where CQ represents the drag coef? cient Rolling Moment: The rolling moment of propeller exists in forward ? ight when the advancing blade is producing more lift than the retreating one. It is the integration over the entire rotor of the lift of each section acting at a given radius. Rm = CRm ? A(? Rrad )2 Rrad 1 1 1 CRm = ? µ ( ? 0 ? ?tw ? ? ) , ?? 6 8 8 where CRm represents the rolling moment coef? cient Ground Effect: is a condition of improved performance encountered when operating near the ground. It is due to the interference of the surface with the air? ow pattern of the rotor system, and it is pronounced the nearer the ground is approached.Increased blade ef? ciency while operating in ground effect is due to two separate and distinct phenomena. The most important is the reduction of the velocity of the induced air? ow illustrated in Figure 2. 3. Since the ground interrupts the air? ow under the helicopter, the entire ? ow is altered. This reduces downward velocity of the induced ? ow. The result is less induced drag and a more vertical lift vector. The lift needed to sustain a hover can be produced with a reduced angle of attack and less power because of the more vertical lift vector.The second phenomena is a reduction of the rotor tip vortex which is also illustrated in Figure 2. 3. (2. 7) Total Aerodynamic Force Lift More Vertical Relative Induced Drag Wind Direction of Airfoil Rotor Tip Reduced Angle of Induced Wind Velocity Reduced Rotor Downwash Angle Reduced Reduction of the velocity of the induced airflow Reduction of the rotor tip vortex Figure 2. 3: Induced air? ow velocity reduction and rotor tip vortex reduction due to ground effect A simple model that captures the main reduction of the induced in? ow velocity considers that at constant power TOGE ? i,OGE = TIGE ? ,IGE the velocity induced at the rotor center by its image is ? ?i = A? i /16? z2 , where z is the position in altitude. This is obtained based on the 14 2. 1 Quadrotor Dynamics assumption that ? i and ? ?i are constant over the rotor disk, thus allowing ? i,IGE = ? i ? ? ? i : TIGE TOGE = 1 rad 1 ? 16z2 R2 . (2. 8) Additionally, if the in? ow ratio In Ground Effect (IGE) can be considered as ? IGE = (? i,OGE ? ? ? i ? z)/? Rrad , where the reduction of the induced velocity is ? ?i = ? i /(4z/Rrad 62). ? Rewriting the thrust coef? cient IGE we get: TIGE IGE CT ?? IGE = CT ? A(? R2 ) rad OGE ? ?i CT = + . ? 4? Rrad (2. 9) Although the general rule of facing ground effect states that ground effect becomes noticeable at one rotor radius, in quadrotors has been observed that this phenomenon is of some importance even in one rotor diameter. 2. 1. 1 General Forces and Moments The quadrotor’s motion is governed by a series of force and moments coming from different physical effects. The model utilized to describe and simulate the quadrotor’s dynamics considers the following symbols have been used (where c cos and s sin, h is the vertical distance ? between the propeller and the CoG, Cd is the drag coef? ient at 70% radial station, and Ac is the fuselage area) as shown in Tables 2. 1, thru, 2. 4: Table 2. 1: Rolling Moments Phenomenon body gyro effect propeller gyro effect roll actuators action hub moment due to sideward ? ight rolling moment due to forward ? ight Expression ?? ? ? (Iyy ? Izz ) ? Jr ? ?r la (? T2 + T4 ) h(? 4 Hyi ) i=1 (? 1)i+1 ? 4 Rmxi i=1 2. 1. 2 Equations of Motion Based on the aforementioned representation of the forces and moments acting on the ? ying quadrotor the nonlinear Newton–Euler model of the system takes the following form: 15 2. QUADROTOR MODELING Table 2. 2: Pitching MomentsPhenomenon body gyro effect propeller gyro effect roll actuators action hub moment due to sideward ? ight rolling moment due to forward ? ight Expression ?? ? ? (Izz ? Ixx ) ? Jr ? ?r la (T1 ? T4 ) h(? 4 Hxi ) i=1 (? 1)i+1 ? 4 Rmyi i=1 Table 2. 3: Yawing Moments Phenomenon body gyro effect inertia counter–torque counter–torque balance hub force unbalance in forward ? ight hub force unbalance in sideward ? ight Expression ?? ? ? (Ixx ? Iyy ) Jr ? r (? 1)i ? 4 Qi i=1 la (Hx2 ? Hx4 ) la (? Hy1 + Hy3 ) Table 2. 4: Forces Along Ez –axis Phenomenon actuators action weight Expression c? c? (? 4 Ti ) i=1 ms gTable 2. 5: Forces Along Ex –axis Phenomenon actuators action hub force in Ex –axis friction Expression s? s? + c? s? c? (? 4 Ti ) i=1 4 ? ?i=1 Hxi 1 ?? 2 Cx Ac ? x|x| 16 2. 1 Quadrotor Dynamics Table 2. 6: Forces Along Ey –axis Phenomenon actuators action hub force in Ey –axis friction Expression ? c? s? + s? s? c? (? 4 Ti ) i=1 ? ?4 Hyi i=1 1 ?? Cy Ac ? y|y| 2 ? ? Ixx ? ? ? ? ? ? ? ? ? ? ? ? ? ?Iyy ? ? ? ? ? ? Jr ? ?r + la (T1 ? T3 ) + h(? r Hxi ) + (? 1)i+1 ? 4 Rmyi ? ? ?? ? i=1 i=1 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Izz ? ? ?? ? + Jr ? r + (? 1)i+1 ? 4 Qi + la (Hx2 ? Hx4 ) + la (? Hy1 + Hy3 )? ?? ? i=1 ? ? ? ? ? ? ? ? (2. 10) ? ? = ? ? ? ? ? 4 ? ms z ? ? ? ms g ? (c? c? ) ? i=1 Ti ? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? 4 4 1 ? ? ms x ? ? (s? s? + c? s? c? ) ? i=1 Ti ? ?i=1 Hxi ? 2 Cx Ac ? x|x| ?? ? ?? ? ? ? ? ? ? ? ? ? ? 4 4 1 ?? ms y ? (? c? s? + s? s? c? ) ? i=1 Ti ? ?i=1 Hyi ? 2 Cy Ac ? y|y| ? ? ? ?? ? ? + Jr ? ?r + la (? T2 + T4 ) ? h(? r Hyi ) + (? 1)i+1 ? 4 Rmxi i=1 i=1 ? This modeling of the quadrotor’s nonlinear 6–Degrees of Freedom (DOF) equations were utilized in order to develop a precise simulation model of the system. 17 2. QUADROTOR MODELING 2. 2 Piecewise Af? ne Modeling ApproachThe main control approaches proposed within this thesis is based on a new modeling approach for the quadrotor’s linearized dynamics. Speci? cally, a family of Piecewise Af? ne (PWA) representations was derived in order to produce piecewise linear systems that cover a large part of the system’s ? ight envelope while also taking into account the additive disturbance effects of atmospheric turbulence as af? ne terms. By utilizing the Euler–Lagrange formulation, the simpli? ed nonlinear model (the hub and friction forces and the rolling moments are neglected) can be described by the following set of twelve ODEs of the following form: ? X = f (X, U) + W , (2. 11) with U ? ?5 the input vector, and X ? ?12 the state vector that consists of: a) the translational ? ? ? ? components and their derivatives, de? ned as: ? = [x, y, z]T , ? = [x, y, z]T , and b) the rotational components and their derivatives, with respect to the ground, de? ned by the vectors: ? = ? ? ? ? ? ? [? , ? , ? ]T , ? = [? , ? , ? ]T , with ? , ? , ? , ? ? ? 3 . The effects of the external disturbances ? in equation (2. 11), have been considered by the additive disturbance vector W ? ?12 , de? ned as: ? ? ? ? ? ? ? ? ? W = [W? | W? ]T = [0, W1 , 0, W2 , 0, W3 | 0, W4 , 0, W5 , 0, W6 ] (2. 2) ? ? where W? and W? are the external additive disturbance vectors, acting on the translational and rotational motions respectively. Equation (2. 11) in its augmented form can be stated as in (3): ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ?? ? ? ? ?? ? ? ?? ? ?? ? ? ? ? ? ?? ? ? ? ?? ? ? ? ? ? ?? ? ? ? ? ? ? X = ? ?=? ? ? ? ? ? ? ? ? z? ? ? ? ? ?z? ? ? ?? ? ? ? ? ? ? x? ? ? ? ? ?x? ? ? ?? ? ? ? ? ? ? y? ? y ? ? ? ?? ? ? ? a1 + ? a2 ? r + b1U2 ? ? ? ?? ? ? a3 ? ? a4 ? r + b2U3 ? ? ?? ? ? a5 + b3U4 ? z ? g ? (cos ? cos ? )U1 /ms x ? uxU1 /ms y ? uyU1 /ms ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? W? ? ? ? ? +? ? ? ?? W ? ? ? ? ? ? ? ? ? ? ? (2. 13) 18 2. 2 Piecewise Af? ne Modeling Approach , ? ? ? ? b(? 2 + ? 2 + ? 2 + ? 2 ) U1 1 2 3 4 ? ? ? ? ? b(?? 2 + ? 2 ) ? U2 ? ? 2 4 ? ? ? ? 2 ? ?2 ) ? U = ? U3 ? = ? b(? 1 3 ? ? ? ? ? ? ? 2 + ? 2 ? ?2 + ? 2 ) ? U4 ? ?d(?? 1 ? 2 3 4 ? ?r ?? 1 + ? 2 ? ?3 + ? 4 ux uy = cos ? sin ? cos ? + sin ? sin ? cos ? sin ? sin ? ? sin ? cos ? (2. 14) (2. 15) a1 = (Iyy ? Izz )/Ixx a2 = Jr /Ixx a3 = (Izz ? Ixx )/Iyy . a4 = Jr /Iyy a5 = (Ixx ? Iyy )/Izz b1 = la /Ixx b2 = la /Iyy b3 = 1/Izz (2. 16) The input U1 ? ? is related to the total thrust, the inputs U2 , U3 , U4 ? are related to the rotations of the quadrotor, ? r ? ? is the overall residual propeller angular speed, ? 1 , · · · , ? 4 ? ?, are the propellers’ angular speeds respectively, m is the total mass of the quadrotor, and g = 9. 81m/sec2 is the gravitational acceleration. Parameters ux and uy represent the directions of thrust vector that cause the motion about the Ex and Ey axis respectively and U1 , in combination with the ? and ? states, rule the altitude motion. The rest of the parameters in equations (2. 132. 16) are listed in Table 2. 7. Table 2. 7: Quadrotor Model Parameters Ixx Iyy Izz la b d JrMoment of Inertia of the quadrotor about the Ex axis Moment of Inertia of the quadrotor about the Ey axis Moment of Inertia of the quadrotor about the Ez axis Quadrotor’s Arm length Thrust coef? cient Drag coef? cient Moment of inertia of the rotor about its axis of rotation Since the angles ? , ? , ? are independent of the translational–vector component, the aforementioned system’s attitude dynamics in equation (2. 13), can be decoupled from the translational ones, an assumption valid for small velocities as explained in (3, 4). Additionally, as 19 2. QUADROTOR MODELING depicted in (2. 3), the altitude and horizontal motions can also be decoupled. Under these considerations, three system models are being derived in order to mathematically describe the vertical, horizontal and rotational motions of the quadrotor as shown in Figure 2. 4. These system models, represent the complete quadrotor, as a set of cascade interconnected systems. The altitude subsystem is related to the total thrust, the horizontal x ? y motion subsystem commands the horizontal position in Ex , Ey axis and produces the reference commands to the rotational subsystem which commands the attitude of the quadrotor. Vertical Motion z,z Rotational Motions ? ? ,? ,? ,? ,? Horizontal Motions x,x,y,y Figure 2. 4: Subsystems of quadrotor’s dynamics 2. 2. 1 Attitude Dynamics In order to derive the Piecewise Af? ne representation of the quadrotor’s attitude dynamics, ? ? ? small attitude perturbations ?? , with ? ? Z+ , around the operating points [0, ? ?,? , 0, ? ?,? , 0, ? ?,? ]T are being assumed, and thus equation (2. 13), only for the attitude behavior, results in: ? ? ?,? ? ? ?,? ?? ? ? ,? ? ?? ? ? ? ? ,? ?? ? ? ,? ?? ? ? ? ? ,? ? ? + ? ??? ? +??? + ? ??? ? ? ? ? ,? + ? ??? ? ? ?,? ? ?.? ? ? ? ? ? ? ?,? a2 ? o + b1U o + ? ? ? ? ? ,? a1 + ? ? ? a2 ? o + b1 ? U2 + ? ?,? ? ? ? a1 + ? ,? ? ? ? a1 + ? ?,? a2 ? ?r + ? ? ? ? ? ? a1 + ? ? ? a2 ? ?r + W2 +? W2 ? ? ? ? ? ? ? ? ? ? ? a1 + ? r r 2 ? ? ? ? ? ? ? ,? + ? ??? ? ? ?? ?,? ? ?.? a + ? ?,? a ? o + b U o + ? ? ? ? ? ,? a + ? ? ? a ? o + b ? U + ? ?,? ? ? ? a + ? ?,? ? ? ? a + ? ?,? a ? ? + ? ? ? ? ? ? a + ? ? ? a ? ? + W + ? W ? . ? 4 r ? ? ? 4 r ? ? ? ? ? ?4 ? 4 ? ? ? 3 r ?? 3 4 r 2 3 3 2 3 3 3 4 ? ? ? ? ? ? ? ,? + ? ??? ? ? ,? ? ? ,? a5 + b3U o + ? ? ? ? ? ,? a1 + b3 ? U4 + ? ?,? ? ? ? a5 + ? ?,? ? ? ? a5 + ? ? ? ? ? ? a5 + W6 + ? W6 ? ? ? ? ? ? ? ? ? ? 4 ? ? ? ? ?= ? ? +??? ? ? + ? ??? ? ? + ? ?? ? (2. 17) 20 2. 2 Piecewise Af? ne Modeling Approach ? ?Neglecting the higher order dynamics (i. e. ? ? ? ? ? ) equation (2. 17) results in: ? ? ? 0 1 ??? ? ? ?? ? 0 ? 0 ?? ? ? ? ? ?? ? ? 0 ?? ? ? 0 ? ? x=? ?? = ? ?0 ? 0,? a ? ? ? ?? ? 3 ? ? ? ? ?? ? ? 0 ?? ? ? 0 ? ? ? ?? 0 ? 0,? a5 ? 0 0 0 ? ? 0 b1 0 ? ?0 0 0 ? ?0 0 b 2 ? ? ? 0 0 0 0 0 0 ? 0 0 0,? a ? 0 ? 1 0 1 0 0 0 0 ? 0 ? 0,? a5 0 0 0 0 0 b3 ?? ? ? 0 0 ?? ? 0,? a1 ? ? ? ? ? ? 0 ? ?? ? ? ?? ? 0 0 ? ?? ? ? ? ?? ?+ ? 0 ? 0,? a3 ? ? ? ? ? ? ?? ? ? ?? ? 0 1 ? ?? ? ? ? ? 0 0 ? ?? ? ? ? ? ? 0 0 ? ? ? U1 ? ?? W2 ? a2 ? ?,? ? ? ? ? ? ? ? ? U2 ? ? ? ? ? ? 0 ?? ? ? ? U3 ? + ? 0 ? . ? ?? W ? ? ? ,? ? ? ?4 ? a4 ? ? ? ? ? ? ? ? U4 ? ? ? ? 0 ? 0 ? ? r ? 0 ? W6 (2. 18) Augmenting this representation with the integral terms ? , ? , ? angles in order to achieve accurate, zero steady–state error setpoint control the following PWA representation can be derived: ? x? x? u? ? ? ? = A? x? + B? u? + W? (2. 19) ? ? dt, ? , ? , = ? ? ,? , ? ? dt, ? , ? , ? dt T = [? U1 , ? U2 , ? U3 , ? U4 , ? ?r ]T ? W? = = [0, ? W1 , 0, 0, ? W2 , 0, 0, ? W3 , 0]T ? A? ? ? ? ? ? ? ? ? = ? ? ? ? 0 ? ? 0 ? ? ? 0 0 ? 0 0 1 0 0 1 0 0 0 Izz ? Ixx ? ,? ? ? I yy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 Iyy ? Izz ? ,? ? ? I xx 0 1 0 0 0 Ixx ? Iyy ? ?,? Izz ? 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 Iyy ? Izz ? ,? Ixx ? 0 0 Iyy ? Ixx ? ?,? Iyy ? 0 0 Ixx ? Iyy ? ?,? Izz ? 0 0 0 0 0 0 ? ? ? ? ? ? ? ? ? ? ? 0 ? ? 0 ? ? ? 0 ? 0 0 0 0 0 0 ? (2. 20) 21 2. QUADROTOR MODELING ? B? ? where W? is the vector that represents the additive external disturbances, affecting the rotational movements of the quadrotor. Let a Ts? the sampling period, equation (2. 19) can be discretized in order to compute the switching control laws, with ? the switching rule: ?? ?? ? x? (k + 1) = A? x? (k) + B? u? (k) + w? , (2. 22) ? ? ? ? ? ? ? ? = ? ? ? ? 0 ? ? 0 ? ? 0 ? 0 ? 0 0 0 0 0 0 la Ixx 0 0 0 0 0 0 0 0 0 0 0 la Iyy 0 0 0 0 0 0 0 1 Izz 0 Jr ? ?,? Ixx ? 0 0 Jr ? ?,? Iyy ? 0 0 0 0 0 0 0 0 0 ? ? ? ? ? ? ? ? ?, ? ? ? ? ? ? ? ? (2. 21) ?? ?? ? ? where x? (k), A? , B? , u? (k), w? the discrete time versions of x? , A? , B? , u? , W? respectively. ? ? 2. 2. 2 Translational Dynamics By transforming the altitude dynamics in equation (2. 13) into error dynamics and discretizing with a Tst ? ?+ sampling time, the following PWA state space representation (xEz = ? ? [? (t), z(t), z(t)dt]T = [z ? zr , z ? zr , (z ? zr )dt]T ) can be extracted, as in (5), where the inz ? ? ? + denotes the reference values for the corresponding variables: dex r ? Z ? ?E ? xEz (k + 1) = AEz xEz (k) + Bv z uEz (k) + wEz , ? 1 Tst 0 ? ? ? AEz = ? 0 1 0? Tst 0 1 ? ? 0 ? Tt ? ?E Bv z = ? mss cos ? ?,v cos ? ?,v ? 0 uEz = [? U1 ] (2. 23) (2. 24) (2. 25) (2. 26) ?E It should be noted that only Bv z depends on the selected nominal operation points ? ?,v and ? ?,v used for the derivation of equation 2. 23. Finally, the system model for the altitude of the quadrotor, in equation (2. 23), can be cast as a PWA system, with v ? Z+ the switching rule. For the horizontal error dynamics, let 22 2. 3 Aerodynamics Modeling T T ? xEx Ey = x(t), x(t), ? ?

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